Talk:Tangent space

Formulas

Some formulas in the latter part of the article appear to be written in LaTEX, but they don't parse! —Preceding unsigned comment added by 217.127.9.138 (talk) 22:32, 4 September 2008 (UTC)

Everything seems to render correctly for me. Sometimes LaTeX formulas, which render as PNG graphics, don't get cached right away if the server load is high, and that might be the problem. At any rate, give it a few hours, and it should be ok. siℓℓy rabbit (talk) 22:46, 4 September 2008 (UTC)

"Tangent map"

The term "tangent map" appears in the Wiki list of missing math topics. Could somebody familiar with this article "Tangent space" please work it in? TIA --LDH (talk) 14:18, 28 November 2008 (UTC)

Where is this formula?

I didn't really read the whole article, but I noticed that it doesn't include the formula fx(x0,y0)(x-x0)+fy(x0,y0)(y-y0)-(z-z0) = 0. Is there a reason for this? --BiT (talk) 15:32, 28 April 2009 (UTC)

this ought to be included. I was just about to say the same thing. BriEnBest (talk) 00:23, 16 November 2010 (UTC) ... well, at least something similar: n dot (position - point) = zero -> normal vector dot ((x,y,z) - (x₀,y₀,z₀)) = 0 BriEnBest (talk) 00:27, 16 November 2010 (UTC)

Most...Rubbish...Intro...Ever

Please! We *can* do better than this atrocious opening paragraph. Blitterbug (talk) 16:03, 5 December 2010 (UTC)

Applications to graphics programming?

I found this term in an article about "how to be a graphics programmer". Should there be some mention of applications in this article? —Preceding unsigned comment added by 24.99.60.219 (talk) 18:48, 12 May 2011 (UTC)

I found this article which provides an onward reference but I do not see much on Wikipedia about the subject - I guess I'm looking in the wrong place! 188.220.56.222 (talk) 17:01, 19 August 2011 (UTC)

Relation to exponential map for Lie groups?

There is no mention of how this concept relates to the exponential map for Lie groups. My understanding is that it is closely related, but must admit, I'm confused exactly how it works. My sense is that the Lie algebra is the tangent space at the identity, $T_{e}G$, but I don't know how that relates to the tangent space at other group elements. I think it's basically that you can talk about the tangent space to another group element, $g$ by thinking of a tangent vector $v\in T_{g}G$ as though it is in the tangent space of the identity and then defining the exponential map from the tangent space of g as

$\exp _{g}v=(\exp _{e}v)g$ (or is it $g\exp _{e}v$ or $g(\exp _{e}v)g^{-1}$ or something?)

That is, following the geodesic defined by $v\in T_{g}G$ is the same as using the exponential map at the origin and then "transporting" that transform over to g. Is that right? Which version is right? I think this also has to do with using a group automorphism to "use" the tangent space at the origin everywhere. Apologies for the vagueness. —Ben FrantzDale (talk) 15:52, 9 June 2011 (UTC)

Twice the dimension? Is that related to Jolt cola having twice the caffiene?

The article says "all the tangent spaces of a manifold form another manifold of twice the dimension"

Twice the dimension of WHAT? The original manifold? If so, please add that.

Question that doesn't belong on the talk page but I'm asking anyway because I don't give a rats ass in hell about any rule that limits my understanding:

Okay, a sphere has dimension two and is embedded in 3-space. Every tangent space of that sphere is a plane. Combining them all gives you the universe, minus the inside of the sphere, and is called the tangent bundle. It has dimension 3. In what sense does the tangent bundle have "twice the dimension" of something? HelviticaBold 07:42, 27 February 2012 (UTC)

You're right, there is room for improvement in this article. Something that isn't made perfectly clear is that these tangent spaces, as described in this article, are treated as abstract manifolds. For your sphere embedded in R^3, the tangent planes are not considered as subsets of the same R^3, but as vector spaces that have an independent existence. So the tangent bundle of a two-dimensional sphere doesn't live inside 3-space; it really is a four dimensional manifold (a two-dimensional family of two-dimensional planes; two plus two makes four). Jowa fan (talk) 08:27, 27 February 2012 (UTC)

This example was very helpful. Could it be inserted into the article? — Preceding unsigned comment added by Marmentad (talk • contribs) 14:58, 17 April 2013 (UTC)

The tangent space at x is not the vector space of derivations at x of smooth functions

It is the vector space of derivations of the algebra of germs of smooth functions at x. — Preceding unsigned comment added by 68.193.47.165 (talk) 19:09, 23 September 2012 (UTC)

Actually in the smooth setting, the definition via derivations at x of smooth functions on the full manifold does work, although it may not be the most elegant way of doing things. 192.52.1.52 (talk) 09:27, 10 October 2017 (UTC)

Examples, please!!!

We are not all mathematicians, you know. — Preceding unsigned comment added by Koitus~nlwiki (talk • contribs) 03:16, 14 December 2019 (UTC)

Edit the Definition via derivations section?

In the article, a derivation at $x\in M$ is defined to be a linear map $D\colon C^{\infty }(M)\to \mathbb {R}$ satisfying the Leibniz rule $D(fg)=D(f)\cdot g+f\cdot D(g)$.

I think that we should edit this to $D(fg)=D(f)\cdot g(x)+f(x)\cdot D(g)$ for the sake of precision, since the definition should obviously involve $x$. I tried to make this edit myself, but it was revoked. — Preceding unsigned comment added by SFeesh (talk • contribs) 00:53, 19 May 2021 (UTC)

You are absolutely right. Derivation takes a function and returns a value. I'm glad you pointed out the error. There was this problematic edit I did not notice earlier because WP did not notify me (until now). So, that's a WP problem, too. StrokeOfMidnight (talk) 02:39, 20 May 2021 (UTC)

Question, isn’t this a derivative at a point instead of a derivation? Sorry to be pedantic but from what I understand of derivation, the codomain here isn’t the family of smooth functions and the reals aren’t a bimodule over the ring of smooth functions. Kurianw (talk) 18:42, 20 September 2023 (UTC)